Semiclassical spectral analysis of Toeplitz operators on symplectic manifolds: the case of discrete wells
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Mathematische Zeitschrift
Semiclassical spectral analysis of Toeplitz operators on symplectic manifolds: the case of discrete wells Yuri A. Kordyukov1 Received: 27 September 2018 / Accepted: 20 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We consider Toeplitz operators associated with the renormalized Bochner Laplacian on high tensor powers of a positive line bundle on a compact symplectic manifold. We study the asymptotic behavior, in the semiclassical limit, of low-lying eigenvalues and the corresponding eigensections of a self-adjoint Toeplitz operator under assumption that its principal symbol has a non-degenerate minimum with discrete wells. As an application, we prove upper bounds for low-lying eigenvalues of the Bochner Laplacian in the semiclassical limit. Keywords Bochner Laplacian · Symplectic manifolds · Semiclassical analysis · Berezin-Toeplitz quantization · Eigenvalue asymptotics Mathematics Subject Classification Primary 58J50; Secondary 53D50 · 58J37
1 Preliminaries and main results 1.1 Introduction The Berezin-Toeplitz operator quantization is a quantization method for a compact quantizable symplectic manifold, which is a particularly effective version of geometric quantization. It was Berezin who recognized the importance of Toeplitz operators for quantization of Kähler manifolds in his pioneering work [4]. There are several approaches to Berezin-Toeplitz and geometric quantization (see, for instance, survey papers [1,16,33,41]). For a general compact Kähler manifold, the Berezin-Toeplitz quantization was constructed by BordemannMeinrenken-Schlichenmaier [5], using the theory of Toeplitz structures of Boutet de Monvel and Guillemin [8]. In this case, the quantum space is the space of holomorphic sections of tensor powers of the prequantum line bundle over the Kähler manifold. In order to generalize the Berezin-Toeplitz quantization to arbitrary symplectic manifolds, one has to find a
Supported by the Russian Science Foundation, project no. 17-11-01004.
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Yuri A. Kordyukov [email protected] Institute of Mathematics with Computing Centre, Ufa Federal Research Centre of Russian Academy of Sciences, 112 Chernyshevsky str., 450008 Ufa, Russia
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substitute for this quantum space. A natural candidate suggested by Guillemin and Vergne is the kernel of the spinc Dirac operator. The Berezin-Toeplitz quantization with such a quantum space was developed by Ma-Marinescu [35,37]. It is based on the asymptotic expansion of the Bergman kernel outside the diagonal obtained by Dai-Liu-Ma [13]. Another candidate suggested by Guillemin-Uribe [17] is the space of eigensections of the renormalized Bochner Laplacian corresponding to eigenvalues localized near the origin. In this case, the Berezin-Toeplitz quantization was recently constructed in [26,28], based on Ma-Marinescu work: the Bergman kernel expansion from [36] and Toeplitz calculus developed in [37] for spinc Dirac operator and Kähler case (also with an auxiliary bundle). We note also tha
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